People are always talking about conserving energy, to save money, and help the environment. They're quite wrong about there being any energy shortage. There's TONS of energy around you.

At 22 Degrees Celsius, the internal energy (that is, the amount of energy that air has stored in it as a function of temperature and pressure) of air is 210.49 kilo joules per kilogram. This is about half as much as water has, just before it boils. Thus, an average sized room probably has enough energy stored up in it to run your computer for a few seconds. Ok, this may not seem like much, but if you look at something like the atmosphere, there's a *lot* of energy in there.

So why don't we all run our cars on air? We've got a lot of it. The problem is that there's really no way to get that energy *out* of the air.

You see, back when they were just figuring out the laws of thermodynamics, two guys figured out that in order to run a cycle that produces work, you cannot run it off a single heat source, without expelling waste heat elsewhere.

What you have to do is take that energy, use some of it, and then expel the rest of it. For example, a steam turbine takes hot steam at high pressure, runs it through the turbine, where the steam transfers a lot of its energy out as work, and then it needs to run the colder, low pressure steam through a condenser, to return it to water. Then it can once again go through the compressor and the boiler needed to change it back into high pressure, hot steam.

While it's going through the condenser, it is transferring a great deal of energy out to the surroundings, and this energy is usually just dissipated into the atmosphere, or into a lake or river. Once in a while, you're able to use it to heat a building, or something along those lines.

But wait a minute, you may say, what about a heat engine? What if we stick a heat engine in between the source of the waste heat, and the surroundings? We can use that to harness some of the waste energy!

Some of it. One of the first forms of the 2nd law of thermodynamics basically stated that work can be transformed directly into heat, but heat cannot be transformed directly into work. Well, if we take that superheated steam, and instead of running it through a reversible heat engine, such as a Carnot engine, and run it until the steam reaches the same temperature and pressure as its surroundings, then we'll have sucked every bit of work out of that steam as we possibly could.

And that's pretty much the definition of exergy. Remember? That's what this node is about. Exergy is the maximum possible work that a specified system (or state) could deliver without breaking any of the laws of thermodynamics.

So, when calculating exergy, we pretend that stuff like friction doesn't exist. We pretend that there isn't any heat transfer to the surroundings when it's not wanted. Does this stuff actually happen? Yes, but since exergy is a theoretical maximum, not the actual amount, it doesn't matter. The idea here is that you want to try and get your actual amount as close to the theoretical amount as possible, by reducing these factors.

Anyways, if you have a moving system, the kinetic energy of that system can, in theory, be converted 100% into work. Same goes for a system with gravitational potential energy. So, a turbine in a hydroelectric dam doesn't really need to be concerned with exergy. Where we do worry about exergy, however, is when we have something that's expected to do work because it's hot, or at a different pressure than the surroundings.

If you have a piston, or some other fixed mass system that can involve a volume change, some of the work will be applied to pushing on whatever the piston is attached to, but some of the work must be used to push back the atmosphere. The work needed to displace a fluid is equal to the pressure of the fluid, times the volume change. On a per unit mass basis,

∂wfrom system = ∂wuseful + P0∂v

As for some sort of a flowing system, such as a turbine, the Volume doesn't change, therefore we don't really need to worry about this.

As for something that's hot transferring energy to its surroundings (also called the dead state, because it can't do any work), well, according to Carnot the maximum thermal efficiency of a heat engine is given by:

Nrev = 1 - TL / TH

Where TL is the absolute temperature of the low temperature reservoir, where the waste heat get expelled to, in this case the temperature of the dead state, and TH is the absolute temperature of the high temperature reservoir, in this case the temperature of the system, at the point we're trying to figure out the exergy of it.

Since this is the ratio of Work from the heat engine, to Heat used,

∂wHE = ( 1 - T0/T)∂q

But, we know from the 2nd law of thermodynamics that ∂s = ∂q/T, ∂q = ∂wHE - T0∂s

Combine this with the first law of thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system, minus the work done by the system, or

∂u = ∂q - ∂w

and add them all together, you get

∂w = -∂u - P0∂v + T0∂s

Integrate, and voila!

wmax = Φ = (u - u0) + P0(v - v0) - T0(s - s0)

or, if you also include possible kinetic or gravitational potential energy,

Φ = (u - u0) + P0(v - v0) - T0(s - s0) + Vel2/2 + gz

And then there's flowing systems. When we're dealing with flowing systems, instead of dealing with the pressure and the internal energy of the system, it's usually easier to deal with the enthalpy of the system. Since h = u + Pv, the above equation simplifies to

Ψ = (h - h0) - T0(s - s0) + Vel2/2 + gz

for steady state flowing systems.

But, it's not really the absolute value that matters all that much. Most of the time you want to be looking at the change in exergy during a process. For a closed system this is

ΔΦ = (u2 - u1) + P0(v2 - v1) - T0(s2 - s1) + (Vel22 - Vel12)/2 + g(z2 - z2)

and for a flowing system

Ψ = (h2 - h1) - T0(s2 - s1) + (Vel22 - Vel12)/2 + g(z2 - z1)

Finally, we're getting to something useful. Now we can find the 2nd law efficiency of a process. Take any process whatsoever, and there will be some exergy supplied to the process, and some exergy recovered from the process. For example, in a turbine, the exergy supplied will be the exergy change of whatever is powering the turbine, whereas the exergy recovered is the actual work output of the turbine. Or for a compressor, the exergy supplied is the actual work needed to run the compressor, whereas the exergy recovered is the amount of work needed to compress the gas in a reversible manner, in other words, the minimum theoretical work needed to run the device.

This gives us two efficiencies. The thermal efficiency is the ratio of output to input. e.g. The amount of work you get out of a steam turbine compared to the amount of heat supplied to the steam by the boiler. On the other hand, the 2nd law efficiency is the ratio of the work done to the work that *could* be done, given the same start and end states. It is that ratio of the thermal efficiency to the maximum thermal efficiency.

This is in many ways a much more useful quantity. After all, it may sound bad when a power plant is only running at 60% thermal efficiency, but when you go on to explain that this is at 92% 2nd law efficiency, it really doesn't look all that bad.

But how do we figure out how much exergy is wasted in any process? Well, take an isolated system, large enough to include the surroundings. We know that the exergy change of this will be

ΔΦ = (u2 - u1) + P0(v2 - v1) - T0(s2 - s1) + (Vel22 - Vel12)/2 + g(z2 - z2)

But since it's an isolated system, the average internal energy cannot change, so (u2 - u1) = 0, the volume stays constant, so P0(v2 - v1) = 0, and the system won't have any net velocity or potential energy changes. Change it from per unit mass to total, and it turns out that the exergy destroyed is equal to the dead state temperature times the entropy generated.

ΔXdest = T0Sgen

How useful is this? Well, oftentimes it's hard to pin down exactly how much entropy is ever generated. However, it's usually a fair bit easier to calculate the wasted exergy, and from that it's simple to find the entropy generated.

But how useful is ANY of this? In the end it works down to economics. If we had an unlimited amount of stuff that we could burn to get energy, then we probably wouldn't care too much about how efficient we are. But, unfortunately, we are running out, so we want to squeeze out every drop of usable work as we can. Exergy is basically a tool that allows us to measure how close we are to doing so.

Sources: Thermodynamics: An Engineering Approach, Cengel & Boles, 4th edition, 2001.
ENME485 at the University of Calgary